Convergence and learning curve of 2nd order LMS predictor

** lab15_2.m * mw * 05/19/2007

Contents

2nd order model system, optimum prediction coefficients, MMSE

a = [1 -.71 .25];        % denominator coefficients
b = .24192*[1 2 1];      % numerator coefficients
Nh = 21;                 % lenght of impulse response
h = impz(b,a,Nh);        % impulse response
Rhh = conv(h,flipud(h)); % acf
R = [Rhh(Nh) Rhh(Nh+1);
     Rhh(Nh+1) Rhh(Nh)]/Rhh(Nh);     % correlation matrix
r = [Rhh(Nh+1); Rhh(Nh+2)]/Rhh(Nh);  % crosscorrelation vector
fprintf('lab15_2 mw 05/19/2007\n\n')
fprintf('2nd order correlation matrix\n')
fprintf('Rxx[1,1] = %6.4f ; Rxx[1,2] = %6.4f \n',R(1,1),R(1,2))
fprintf('Rxx[2,1] = %6.4f ; Rxx[2,2] = %6.4f \n',R(2,1),R(2,2))
fprintf('Crosscorrelation vector\n')
fprintf('rxd[1] = %6.4f ; rxd[2] = %6.4f \n',r(1),r(2))
b_opt = R\r;                         % MMSE design
MSEopt = R(1,1) + b_opt'*R*b_opt - 2*b_opt'*r; % minimum MSE
fprintf('Prediction coefficients and error signal power\n')
fprintf('b0 = %6.4f ; b1 = %6.4f ; MSEopt = %6.4f\n',b_opt(1),b_opt(2),MSEopt)
lambda = eig(R); % eigenvalues of the correlation matrix
fprintf('Eigenvalues\n')
fprintf('lambda1 = %6.4f ; lambda2 = %6.4f \n',lambda(1),lambda(2))

lab15_2 mw 05/19/2007

2nd order correlation matrix
Rxx[1,1] = 1.0000 ; Rxx[1,2] = 0.7885 
Rxx[2,1] = 0.7885 ; Rxx[2,2] = 1.0000 
Crosscorrelation vector
rxd[1] = 0.7885 ; rxd[2] = 0.3684 
Prediction coefficients and error signal power
b0 = 1.3168 ; b1 = -0.6700 ; MSEopt = 0.2085
Eigenvalues
lambda1 = 0.2115 ; lambda2 = 1.7885 

Sample function of model process

M = 1e5;            % number of iterations
randn('state',0);
w = randn(1,M);     % innovation
d = filter(b,a,w);  % IIR filter for process modeling - desired signal

Simulation of adaptiv system

.alpha - convergence parameter, .beta - weighting parameter of variance estimator .sig2 - variance, .N - FIR filter order (N+1 coefficients) .Mode - mode for updating FIR filter coefficients, Mode = 'e*x' no sign function used Mode = 'x*sign(e)','e*sign(x)' and 'sign(e*x)' uses sign function

p = 2;  % prediction order (FIR filter order + 1)
lms = struct('alpha',1/32,'beta',1/64,'b',b_opt,'e',0,'x',zeros(p,1),...
             'sig2',p,'Mode','e*x');
x = [0 d(1:M-1)];   % input signal for adaptive FIR system
% noise = 10^(-10/10)*randn(1,M); % AWGN
[y,e,lms,SAVEb] = lms_algorithm(x,d,lms); % LMS algorithm
fprintf('final prediction coefficients\n')
fprintf('(%i) b0 = %6.4f ; b1 = %6.4f\n',M,lms.b(1),lms.b(2))
EccessMSE = sum(e.^2)/M - MSEopt;
fprintf('Misadjustment = %6.4f\n\n',EccessMSE/MSEopt)

final prediction coefficients
(100000) b0 = 1.3962 ; b1 = -0.6680
Misadjustment = 0.0136

Graphics

FIG1 = figure('Name',['lab15_2 : Desired signal, prediction error and variance estimate (',lms.Mode,')'],...
    'NumberTitle','off');
subplot(3,1,1),plot(0:M-1,d,'LineWidth',2), grid, axis([0 M-1 -4 4])
xlabel('n \rightarrow'), ylabel('d[n] \rightarrow')
title('Desired signal')
subplot(3,1,2),plot(0:M-1,e,'LineWidth',2), grid, axis([0 M-1 -4 4])
xlabel('n \rightarrow'), ylabel('e[n] \rightarrow')
title('Error')
subplot(3,1,3),plot(0:M-1,SAVEb(1,:)/2,'LineWidth',2), grid, axis([0 M-1 0 2])
xlabel('n \rightarrow'), ylabel('\sigma^2_x[n] \rightarrow')
title('Variance estimates')
% MSE learning curve
b = SAVEb(2:2+p-1,:); LC = zeros(1,M);
for n = 1:M
LC(n) = R(1,1) + b(:,n)'*R*b(:,n) - 2*b(:,n)'*r;   % MSE, learning curve
end
FIG2 = figure('Name',['lab15_2 : Learning curve (',lms.Mode,')'],'NumberTitle','off');
plot(1:M-1,LC(1:M-1)/MSEopt,'LineWidth',2), grid
axis([1 M 1 5]);
xlabel('n \rightarrow'), ylabel('MSE[n] / MSEopt \rightarrow')
title('Learning curve (MSE)')
% 2nd order prediction error - MSE
c0 = -2:.05:2; c1 = -2:.05:2;
MSE = zeros(length(c0),length(c1));
for k=1:length(c0)
    for l=1:length(c1)
        cc = [c0(k); c1(l)] + b_opt;
        MSE(l,k) = R(1,1) + cc'*R*cc - 2*cc'*r;
    end
end
FIG3 = figure('Name',['lab15_2 : Realtiv MSE and convergence of coefficients c[n] (',lms.Mode,')'],...
    'NumberTitle','off');
c = b - repmat(b_opt,1,M);
V = [1 1.1 1.2 1.5 2 3 4 6];
[cc,h] = contour(c0,c1,MSE/MSEopt,V,'LineWidth',1,'Color','b'); grid
clabel(cc);
xlabel('c_0 \rightarrow'), ylabel('c_1 \rightarrow')
title('Convergence of coefficients and MSE')
hold on
plot(c(1,:),c(2,:),'o','MarkerSize',4,'MarkerFaceColor','b')
hold off

Published with MATLAB® 7.3